Risk and Returns: Simulating Returns


Kerry Back

BUSI 721, Fall 2022
JGSB, Rice University

Is there risk in the long run?

A common misconception is that the “law of averages” will rule in the long run, eliminating risk.

It is true that the average return should not be risky in the (very) long run.

But gambling many times does not eliminate risk. The average gain per gamble may equal the true odds in the long run, but the gain or loss is

average gain per gamble \(\times\) number of gambles.

Let’s simulate normally distributed returns

  • A single history of compounded returns
  • Multiple histories to evaluate the distribution
  • Form predictors from past history to predict the future
  • Retirement planning

Simulating a Single History

Note the variation in compounded returns in different simulations

HTML tutorial

Simulating Multiple Histories

  • The variation in outcomes is captured by the standard dev.
  • Compounding \(\Rightarrow\) positive skewness
  • Mean outcome \(>\) Median outcome
  • Lower risk (for a given mean) \(\Rightarrow\) higher median

HTML tutorial

Retirement Planning Simulation

Same exercise as in our first module

But generate a random return each month of the investment lifetime

And simulate many lifetimes in this way

BBCX Investment Library

HTML tutorial

If we want to forecast a constant return (not simulate) and base the return on history, should we use the historical arithmetic average or historical geometric average?

Geometric average is better for forecasting future compounded returns.

Arithmetic average is better for forecasting future (arithmetic) average return.

HTML tutorial